International Journal of Astronomy and Astrophysics, 2011, 1, 183-199 doi:10.4236/ijaa.2011.14024 Published Online December 2011 (http://www.SciRP.org/journal/ijaa) An Astrophysical Peek into Einstein’s Static Universe: No Dark Energy Abhas Mitra Theoretical Astrophysics Section, Bhabha Atomic Research Centre, Mumbai, India Homi Bhabha National Institute, Mumbai, India E-mail: [email protected] Received June 26, 2011; revised August 28, 2011; accepted September 13, 2011 Abstract It is shown that in order that the fluid pressure and acceleration are uniform and finite in Einstein’s Static Universe (ESU), , the cosmological constant, is zero. being a fundamental constant, should be the same everywhere including the Friedman model. Independent proofs show that it must be so. Accordingly, the supposed acceleration of the universe and the attendant concept of a “Dark Energy” (DE) could be an illusion; an artifact of explaining cosmological observations in terms of an oversimplified model which is fundamentally inappropriate. Indeed observations show that the actual universe is lumpy and inhomogeneous at the largest scales. Further in order that there is no preferred centre, such an inhomogeneity might be ex- pressed in terms of infinite hierarchial fractals. Also, the recent finding that the Friedman model intrinsically corresponds to zero pressure (and hence zero temperature) in accordance with the fact that an ideal Hubble flow implies no collision, no randomness (Mitra, Astrophys. Sp. Sc., 333,351, 2011) too shows that the Friedman model cannot represent the real universe having pressure, temperature and radiation. Dark Energy might also be an artifact of the neglect of dust absorption of distant Type 1a supernovae coupled with likely evolution of supernovae luminosities or imprecise calibration of cosmic distance ladders or other systemetic errors (White, Rep. Prog. Phys., 70, 883, 2007). In reality, observations may not rule out an inhomogeneous static universe (Ellis, Gen. Rel. Grav. 9, 87, 1978), if the fundamental “constant”s are indeed constant. Keywords: General Relativity, Cosmological Constant, Cosmology, Dark Energy, Fractal Cosmology, Static Universe, Big Bang Theory 1. Introduction hand side (LHS) of the same equation contains much more tangible quantity, Tik , the energy momentum ten- The “cosmological principle” demands that not only does sor of the matter generating the space time structure gik the center of the universe lie everywhere, but also any or Gik . Thus any constraint imposed on gik from fundamental observer must see the universe as isotropic symmetry or other considerations must take its toll on the and homogeneous. This means that, the space must be admissible form of Tik as well. This means that con- “maximally symmetric” [1]. By this consideration, one straints imposed on Gik must be reflected on the Tik of can directly obtain not only the metric for Einstein’s sta- the fluid generating the Gik . This latter physics aspect tic universe (ESU) but the non-static Friedmann-Rober- however may remain hidden in a purely symmetry driven tson-Walker (FRW) metric as well [1,2] without using Eins- mathematical derivation of the cosmological metric tein’s equation Gik=8π T ik at all ( Gc==1). Let us in- which does not invoke Einstein equation at all. And in vert it in the form this paper, we would like to explore the tacit constraints imposed on T by the requirements of the cosmological 1 ik Tik = Gik (1) principle, “there is no preferred center”, or “center lies 8π everywhere”. All the results obtained during this process The Einstein tensor Gik , on the right hand side (RHS) will be also ratified by physical considerations as well. of this equation, comprises only geometric quantities, the In effect we shall be dealing with spherically symmet- structure of space time gik . On the other hand, the left ric static solutions of Einstein equation with the inclusion Copyright © 2011 SciRes. IJAA 184 A. MITRA of a cosmological constant , [3,4]. In recent times order that the timelike worldlines of test particles always several authors have shown keen interest in this impor- remain timelike and no trapped surface is formed, FRW tant problem [5-8]. However such studies mostly focus model too has to be vacuous with =0. Then the real attention on isolated objects or stars where there (1) must universe must be something entirely different from the be a pressure gradient, (2) in general, a density gradient, isotropic, homogeneous and continuous FRW/ESU mod- (3) a natural boundary R= Rb where fluid pressure els. Indeed galaxies and structures are found to be dis- must vanish, pRb =0 and (4) where a discontinuity tributed in discrete, lumpy and inhomogeneous manner in the fluid density might also occur. Further in such even at the largest scales. Nonetheless such matter dis- generalized studies, has not been considered as a tribution can still satisfy the “Copernican Principle” of fixed fundamental constant contrary to what was in Ein- no unique centre if it would form infinite hierarchal stein’s mind when it was first introduced. fractal pattern. At the beginning, however, we focus at- In contrast, in the present study Einstein is a basic tention on the ESU. constant and there is no question of modifying its value according to the needs of a general fluid solution. And of 2. Formulation course, there must not be any natural “boundary” at all for the universe so that Copernican principle remains We start with the assumption of spherical symmetry and valid. It may be recalled that the ancient Ionian philoso- consider a general form a static metric [2] pher “Aristarchos of Samos” had proposed the heliocen- d=dseteR22 d22Rd2 (2) tric theory much before Copernicus and hence we may 222 2 also call this as “Aristarchian Principle”. It will be found where d=d sin d and R is the area coor- that both the metric for a star and ESU have a “coordi- dinate. In particular, in a static universe, the luminosity nate singularity at R = , where R is the luminosity distance turns out to be exactly R [2] distance and is an appropriate integration constant. d=L R (3) This singularity gets propagated in the hydrostatic bal- We also assume the cosmic fluid to be perfect with ance equation as well. And in order to ensure that iso- tropic pressure gradient p does not blow up at this Tpik = uupi k gik (4) singularity, relativistic stars are constructed so that where, is the fluid density, p is the isotropic pres- R > . Here a prime denotes differentiation by R. Ac- i b sure, and u is fluid 4-velocity. As is clear from Equa- cordingly one can almost forget this singularity while tion (1), at the beginning, we do not consider any cos- studying static stars. But for the ESU one has to live with 0 mological constant ; and the 0 component of the this singularity. And in this paper, we would study the Einstein equation reads behavior of physical quantities like pressure and accel- eration despite the presence of this singularity. And the 0 1 1 8πTe0 =8π = 22 (5) conclusion is that ESU has to be vacuous with =0 in RRR order to avoid such singularities anywhere. This can be integrated to yield At the very beginning, let us remind the difference R between a fundamantal constant and a model parameter. e =1 (6) Suppose we are studying a room temperature homoge- R neous static gas and we have obtained some expression where R for pressure p , temperature T and particle number RR=8π 2dR (7) density n . But since pn= kT, in principle, we may 0 obtain a numerical value of the Boltzmann constant k . Here the condition (0) = 0 has been used to ensure Once we assume that k is a fundamental constant and that e is regular at R =0. Thus whether, it is the in- not just a model related parameter, the value of k must terior solution of a star or the static universe, general be independent of time t or position r or any other form of the metric is variable. Thus, if we would be studying a gas in a situa- dR2 tion where the gas may be inhomogeneous or may have d=dset22 R 2 d2 (8) 1 R bulk motion, we must be able to use the same numerical value of k obtained in an ideal static homogeneous Note, as of now, we need not necessarily interpret case. Similarly, here we would study the case of the ()R as related to the observed mass though what we supposed fundamental constant though it will be in the have done looks like finding the interior of a relativistic context of ESU. Therefore, we would expect =0 for star by working out the Schwarzschild interior solution. the dynamic FRW model too. It would be found that in The important difference, however, is that while for a Copyright © 2011 SciRes. IJAA A. MITRA 185 star there is a unique center and a boundary where the to rewrite Equation (14) as density may be discontinuous, for the universe, center is 2222 2 2 d=dsetS dsin d (16) everywhere and there is no boundary, no exterior solu- tion, and no density discontinuity. Further even for a There is clearly a singularity in one of the metric coef- constant density star, there must be a pressure gradient ficients in Equations (12) and (14) at RS= or because there is a unique center. On the other hand, for R =1 or r =1. On the other hand, metric (16) has a an isotropic and homogeneous continuous universe, not similar singularity at =0. But one cannot make any a only p and , but all physically meaningful quanti- priori comment on the nature of such singularities with- ties, all geometrical scalars must the same everywhere.